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Proposition 7.5.2.4. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a (strictly commutative) diagram of $\infty $-categories. Then the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ is an $\infty $-category. Moreover, $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} )$ can be identified with a limit of the diagram

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}} \]

in the $\infty $-category $\operatorname{\mathcal{QC}}$.

Proof. By virtue of Example 5.6.5.6, the functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F} )$ is a covariant transport representation for the cocartesian fibration $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Proposition 7.5.2.4 is therefore a special case of Corollary 7.4.1.9. $\square$